A person's decision on whether to turn in the family member or not would depend on where the person was in Kohlberg's six stages. If the person making the decision is an adult, we will assume that they are operating at either level 5 or level 6.
If the person making the decision is at level 5 of moral development, they will surely turn the family member in. At level 5, people tend to put more weight on the "social contract" and on obeying the laws. They do not think as much about more abstract ideas of justice.
If, however, the person making the decision is at level 6, they will have a much more difficult choice. They will have to look at the reasons behind the family member's actions. They will have to try to decide if higher goals (justice, human dignity, etc) are best served by turning the family member in or by letting them get away with their crime.
Therefore, the level the person has reached in Kohlberg's stages will have a great impact on his or her actions in this hypothetical.